As of December 2017, the ** largest known prime number ** is 2 ^{ 77,232,917 } − 1, a number with 23,249,425 digits. It was found on December 26, 2017 by the Great Internet Mersenne Prime Search (GIMPS).

Euclid proved that there is no largest prime number, and many mathematicians and hobbyists continue to search for large prime numbers.

Many of the largest known primes are Mersenne primes. As of January 2017, the six largest known primes are Mersenne primes. ^{ [2] } The last 16 record primes were Mersenne primes. ^{ [3] } ^{ [3] }

The fast Fourier transform implementation of the Lucas–Lehmer primality test for Mersenne numbers is fast compared to other known primality tests for other kinds of numbers.

## The current record

The record is currently held by 2 ^{ 77,232,917 } − 1 with 23,249,425 digits. ^{ [38] }

## Prizes

The Great Internet Mersenne Prime Search (GIMPS) currently offers a US$3000 research discovery award for participants who download and run their free software and whose computer discovers a new Mersenne prime having fewer than 100 million digits.

There are several prizes offered by the Electronic Frontier Foundation for record primes. ^{ [21] } GIMPS is also coordinating its long-range search efforts for primes of 100 million digits and larger and will split the Electronic Frontier Foundation's US$150,000 prize with a winning participant.

The record passed one million digits in 1999, earning a US$50,000 prize. ^{ [13] } In 2008 the record passed ten million digits, earning a US$100,000 prize and a Cooperative Computing Award from the Electronic Frontier Foundation. ^{ [21] } * Time * called it the 29th top invention of 2008. ^{ [8] } Additional prizes are being offered for the first prime number found with at least one hundred million digits and the first with at least one billion digits. ^{ [21] } Both the US$50,000 and the US$100,000 prizes were won by participation in GIMPS.

## History

The following table lists the progression of the largest known prime number in ascending order. ^{ [3] } Here M _{ n } = 2 ^{ n } − 1 is the Mersenne number with exponent * n * . The longest record-holder known was M _{ 19 } = 524,287, which was the largest known prime for 144 years. Almost no records are known before 1456.

Number | Decimal expansion (only for numbers < 10 ^{ 50 } ) | Digits | Year found | Notes (for larger Mersenne primes, see Mersenne prime) |
---|---|---|---|---|

11 | 11 | 2 | ~1650 BCE | ancient Egyptians (disputed) ^{ [14] } |

7 | 7 | 1 | ~400 BCE | It was known to Philolaus that 7 is a prime ^{ [15] } |

M _{ 7 } | 127 | 3 | ~300 BCE | It was known to Euclid that 127 and 89 are primes ^{ [2] } ^{ [17] } |

M _{ 13 } | 8,191 | 4 | 1456 | Anonymous discovery |

M _{ 17 } | 131,071 | 6 | 1460 | Anonymous discovery |

M _{ 19 } | 524,287 | 6 | 1588 | Found by Pietro Cataldi |

2 32 + 1 641 {displaystyle {tfrac {2^{32}+1}{641}}} | 6,700,417 | 7 | 1732 | Found by Leonhard Euler |

M _{ 31 } | 2,147,483,647 | 10 | 1772 | Found by Leonhard Euler |

2 64 + 1 274177 {displaystyle {tfrac {2^{64}+1}{274177}}} | 67,280,421,310,721 | 14 | 1855 | Found by Thomas Clausen |

M _{ 127 } | 170,141,183,460,469,231,731,687,303,715,884,105,727 | 39 | 1876 | Found by Édouard Lucas |

2 148 + 1 17 {displaystyle {tfrac {2^{148}+1}{17}}} | 20,988,936,657,440,586,486,151,264,256,610,222,593,863,921 | 44 | 1951 | Found by Aimé Ferrier with a mechanical calculator; the largest record not set by computer. |

180×(M _{ 127 } ) ^{ 2 } +1 | 79 | 1951 | Using Cambridge's EDSAC computer | |

M _{ 521 } | 157 | 1952 | ||

M _{ 607 } | 183 | 1952 | ||

M _{ 1279 } | 386 | 1952 | ||

M _{ 2203 } | 664 | 1952 | ||

M _{ 2281 } | 687 | 1952 | ||

M _{ 3217 } | 969 | 1957 | ||

M _{ 4423 } | 1,332 | 1961 | ||

M _{ 9689 } | 2,917 | 1963 | ||

M _{ 9941 } | 2,993 | 1963 | ||

M _{ 11213 } | 3,376 | 1963 | ||

M _{ 19937 } | 6,002 | 1971 | ||

M _{ 21701 } | 6,533 | 1978 | ||

M _{ 23209 } | 6,987 | 1979 | ||

M _{ 44497 } | 13,395 | 1979 | ||

M _{ 86243 } | 25,962 | 1982 | ||

M _{ 132049 } | 39,751 | 1983 | ||

M _{ 216091 } | 65,050 | 1985 | ||

391581×2 ^{ 216193 } −1 | 65,087 | 1989 | ||

M _{ 756839 } | 227,832 | 1992 | ||

M _{ 859433 } | 258,716 | 1994 | ||

M _{ 1257787 } | 378,632 | 1996 | ||

M _{ 1398269 } | 420,921 | 1996 | ||

M _{ 2976221 } | 895,932 | 1997 | ||

M _{ 3021377 } | 909,526 | 1998 | ||

M _{ 6972593 } | 2,098,960 | 1999 | ||

M _{ 13466917 } | 4,053,946 | 2001 | ||

M _{ 20996011 } | 6,320,430 | 2003 | ||

M _{ 24036583 } | 7,235,733 | 2004 | ||

M _{ 25964951 } | 7,816,230 | 2005 | ||

M _{ 30402457 } | 9,152,052 | 2005 | ||

M _{ 32582657 } | 9,808,358 | 2006 | ||

M _{ 43112609 } | 12,978,189 | 2008 | ||

M _{ 57885161 } | 17,425,170 | 2013 | ||

M _{ 74207281 } | 22,338,618 | 2016 |

GIMPS found the thirteen latest records (all of them Mersenne primes) on ordinary computers operated by participants around the world.

## The twenty largest known prime numbers

A list of the 5,000 largest known primes is maintained by Chris K. Caldwell, ^{ [2] } the twenty largest are listed below.

Rank | Number | Discovered | Digits | Ref |
---|---|---|---|---|

1 | 2 ^{ 74,207,281 } – 1 | 2016-01-07 | 22,338,618 | |

2 | 2 ^{ 57,885,161 } – 1 | 2013-01-25 | 17,425,170 | ^{ [19] } |

3 | 2 ^{ 43,112,609 } – 1 | 2008-08-23 | 12,978,189 | ^{ [20] } |

4 | 2 ^{ 42,643,801 } – 1 | 2009-06-04 | 12,837,064 | ^{ [21] } |

5 | 2 ^{ 37,156,667 } – 1 | 2008-09-06 | 11,185,272 | ^{ [20] } |

6 | 2 ^{ 32,582,657 } – 1 | 2006-09-04 | 9,808,358 | ^{ [22] } |

7 | 10223 × 2 ^{ 31,172,165 } + 1 | 2016-10-31 | 9,383,761 | ^{ [23] } |

8 | 2 ^{ 30,402,457 } – 1 | 2005-12-15 | 9,152,052 | ^{ [24] } |

9 | 2 ^{ 25,964,951 } – 1 | 2005-02-18 | 7,816,230 | ^{ [25] } |

10 | 2 ^{ 24,036,583 } – 1 | 2004-05-15 | 7,235,733 | ^{ [26] } |

11 | 2 ^{ 20,996,011 } – 1 | 2003-11-17 | 6,320,430 | ^{ [27] } |

12 | 919,444 ^{ 1,048,576 } + 1 | 2017-08-29 | 6,253,210 | ^{ [28] } |

13 | 168,451 × 2 ^{ 19,375,200 } + 1 | 2017-09-17 | 5,832,522 | ^{ [29] } |

14 | 123,447 ^{ 1,048,576 } − 123,447 ^{ 524,288 } + 1 | 2017-02 | 5,338,805 | ^{ [30] } |

15 | 143,332 ^{ 786,432 } − 143,332 ^{ 393,216 } + 1 | 2017-01 | 4,055,114 | ^{ [31] } |

16 | 2 ^{ 13,466,917 } − 1 | 2001-11-14 | 4,053,946 | ^{ [32] } |

17 | 19249 × 2 ^{ 13,018,586 } + 1 | 2007-05 | 3,918,990 | ^{ [34] } |

18 | 3 × 2 ^{ 11,895,718 } − 1 | 2015-06-23 | 3,580,969 | ^{ [35] } |

19 | 3 × 2 ^{ 11,731,850 } − 1 | 2015-03-13 | 3,531,640 | ^{ [36] } |

20 | 3 × 2 ^{ 11,484,018 } − 1 | 2014-11-22 | 3,457,035 | ^{ [37] } |